Navigating by the Stars and Planets Presented to: Chagrin Valley Astronomical Society September 1, 2012 By Ron Baker, CVAS member (Finding your location from measured altitudes of celestial bodies)

Historical Context Navigators have been able to determine their latitude at sea for centuries. But longitude can only be obtained when the accurate time is known. Without longitude, shipwrecks occurred far too often. On October 22, 1707, a fleet of 21 ships under the command of Admiral Sir Clowdisley Shovell of the Royal Navy were returning from Gibraltar to England. After 12 days at sea, 4 ships struck on the rocks of the Scilly Isles and sank. Nearly 2000 lives were lost. This event in particular focused attention on the need to improve accurate time keeping methods.

What Time Is It? Two competing methods for determining and maintaining accurate time emerged starting in the early 18 th century. Astronomers promoted the Lunar Distance Method which uses the predicted positions of the Moon and other celestial bodies for calculating the accurate time. Clockmakers worked to build chronometers which would retain high accuracy over long intervals even when subjected to the adverse conditions while underway.

Lunar Distance Method Nevil Maskelyne, the 5 th Astronomer Royal, embraced this method to calculate time. The arc distance between the Moon and another celestial body such as a star, planet, or the Sun is measured with a marine sextant. The Greenwich time of the observation is determined by matching the measured lunar distance with the predicted distance found in an almanac prepared by astronomers. The idea is straightforward, but carrying out the observations, and making the necessary corrections, was time consuming for early navigators. Another complication involved the difficulty for astronomers in making accurate predictions of the complex motion of the Moon.

Marine Chronometer Clockmakers faced many challenges in the design of a clock to be used at sea. The clock needed to operate properly on a rolling and pitching deck, in all kinds of weather, and in various temperatures and humidity. In 1764, a chronometer built by John Harrison (H-4) was carried on board the Tartar departing from Portsmouth. It arrived in Barbados after a voyage of 47 days, and was later judged to have kept the time during the voyage with an error of only 39 seconds.

Apparent Time Based on the apparent motion of the Sun. Sundials display apparent time. Greenwich Apparent Time (GAT) reflects the apparent motion of the Sun observed in Greenwich, located on the Prime Meridian. Local Apparent Time (LAT) reflects the apparent motion of the Sun observed from any other location. Mean Time Based on apparent time, but with corrections due to the equation of time. Greenwich Mean Time (GMT) is used in today’s Nautical Almanac and is often the time maintained on marine chronometers.

Dead Reckoning On the open ocean, navigators keep close track of their course and speed. But the effect of winds and ocean currents on position can be difficult factors to estimate. Without regular corrections (or fixes), the position recorded in the ship’s log becomes increasingly inaccurate. Early navigators relied on celestial navigation to correct their position. And the results from the sight reductions were also very useful for evaluating their dead reckoning skills.

Sight Reduction Armed with 2 new ways to maintain accurate time on board, navigators derived various methods for determining their geographic position on the open ocean. These methods require knowing the precise time, but also some way of measuring, calculating, and comparing various angles. Observed angles are based on the angles measured with a sextant. Calculated angles are determined using data contained in the Nautical Almanac and must be reduced using spherical trigonometry.

Measuring Altitudes: Marine Sextant The sextant is an instrument designed to precisely measure angles, and is used primarily to measure the altitude of a celestial body above the sea horizon. Half the light appearing in the instrument’s eyepiece is directed from the celestial body with a mirror fixed to the instrument’s moving arm. The other half of the light in the eyepiece is from the horizon. When the celestial body appears aligned with the horizon, the altitude is simply read from the degree scale and the arc minute drum. A vernier scale helps the observer estimate the altitude to a precision of 0.1 arc minutes.

Calculating Angles: The Nautical Almanac The daily pages of the nautical almanac contain the hourly predictions for a celestial body’s arc distance west of the prime meridian. This is known as the Greenwich Hour Angle (GHA). The declination (Dec) of each celestial body is also included. GHA and Dec can therefore be found for the Sun, Moon, Venus, Mars, Jupiter and Saturn, First Point of Aries, and for the 57 “navigational” stars for any hour throughout the year. The Nautical Almanac was first published in 1766 (for the year 1767), and has been produced in various versions annually up to the present time.

Calculating Angles: The Navigational Triangle The navigator relies on a special triangle, which like all spherical triangles, has 3 vertices and 3 arc angles. The navigational triangle is constructed by placing the vertices at the north celestial pole (N), the assumed position of the observer (AP), and the geographic position (GP) of the celestial body being observed. Once the navigator knows the values for 3 of the 6 angles, spherical trigonometry can be applied to calculate any of the remaining 3 angles.

Finding Position at Sea The Local Apparent Time Method for determining longitude was used in connection with the Local Apparent Noon (LAN) sight commonly used by navigators to determine latitude. The Intercept Method came about somewhat later and is a more sophisticated approach. It’s main advantage is that both latitude and longitude are determined simultaneously. Both methods require the use of a marine sextant to measure altitudes of celestial bodies, data from the Nautical Almanac to determine the bodies geographic position, and trigonometry to calculate specific angles. Both methods require that either GMT or GAT be known.

The Local Apparent Time Method The early navigator would obtain his latitude by observing the Sun as it crossed the local meridian at noon. Time was not needed for this step. The altitude observations typically began just before noon and continued until the measured altitude began to decrease. The Sun’s declination must be used in the latitude calculation. The Sun was also observed several hours before or after noon. GAT was known, so the Sun’s declination could be obtained from the Nautical Almanac. The Local Hour Angle (LHA) of the Sun was solved with trigonometry. When the Sun was west of the meridian (an afternoon sight), longitude was calculated by subtracting LAT (equal to LHA) from GAT.

The Intercept Method This method came into use in the early to mid 19 th century. The method is elegant in its design, and has the great advantage of deriving latitude and longitude simultaneously. The altitude of a celestial body is measured with a marine sextant, usually at twilight, and corrected to observed altitude (Ho). Data from the Nautical Almanac are used to determine the geographic position of the celestial body at the time of the observation. Trigonometry is used to solve for the calculated altitude (Hc). The difference between Ho and Hc, and the calculated true azimuth, are plotted on a nautical chart and the fix is found.

Example Using the Intercept Method Sextant observations of several celestial bodies were made near the Huntington Island lighthouse in South Carolina in June Corrections were made to the sights to arrive at the observed altitudes (Ho). Data from the Nautical Almanac were used to find the geographic positions (GP) the bodies. The assumed position (AP) was placed 15 miles east of the shoreline. Calculated altitudes (Hc) were found with the trigonometric methods described in the navigational triangle graphics. Lines of Position (LOPs) were plotted, and a was fix determined. The Moon and Vega were well placed for observation, but Mercury was far too low in the west to be useful.

Measured Altitudes and Sight Reduction Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury

Measured Altitudes and Sight reduction Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury Sextant altitude (Hs)44° 06.7’38° 44.1’ 8° 17.7’

Measured Altitudes and Sight reduction Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury Sextant altitude (Hs)44° 06.7’38° 44.1’ 8° 17.7’ Corrections: Dip of the horizon - 3.4’ - 3.4’ - 2.9’

Measured Altitudes and Sight reduction Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury Sextant altitude (Hs)44° 06.7’38° 44.1’ 8° 17.7’ Corrections: Dip of the horizon - 3.4’ - 3.4’ - 2.9’ Apparent altitude (Ha)44° 03.3’38° 40.7’8° 14.8’

Measured Altitudes and Sight reduction Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury Sextant altitude (Hs)44° 06.7’38° 44.1’ 8° 17.7’ Corrections: Dip of the horizon - 3.4’ - 3.4’ - 2.9’ Apparent altitude (Ha)44° 03.3’38° 40.7’8° 14.8’ Atmospheric refraction - 1.0’- 1.2’- 6.1’

Measured Altitudes and Sight reduction Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury Sextant altitude (Hs)44° 06.7’38° 44.1’ 8° 17.7’ Corrections: Dip of the horizon - 3.4’ - 3.4’ - 2.9’ Apparent altitude (Ha)44° 03.3’38° 40.7’8° 14.8’ Atmospheric refraction - 1.0’- 1.2’- 6.1’ Parallax42.5’0.0’0.2’

Measured Altitudes and Sight reduction Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury Sextant altitude (Hs)44° 06.7’38° 44.1’ 8° 17.7’ Corrections: Dip of the horizon - 3.4’ - 3.4’ - 2.9’ Apparent altitude (Ha)44° 03.3’38° 40.7’8° 14.8’ Atmospheric refraction - 1.0’- 1.2’- 6.1’ Parallax42.5’0.0’0.2’ Semi-diameter ’ 0.0’ 0.0’

Measured Altitudes and Sight reduction Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury Sextant altitude (Hs)44° 06.7’38° 44.1’ 8° 17.7’ Corrections: Dip of the horizon - 3.4’ - 3.4’ - 2.9’ Apparent altitude (Ha)44° 03.3’38° 40.7’8° 14.8’ Atmospheric refraction - 1.0’- 1.2’- 6.1’ Parallax42.5’0.0’0.2’ Semi-diameter ’ 0.0’ 0.0’ Observed altitude (Ho)44° 28.6’38° 39.5’8° 08.9’

Calculated Altitudes, Intercept and Azimuth Date Time GMT00:59:0701:17:1701:25:49 BodyMoon VegaMercury GHA95° 47.2’16° 31.1’173° 18.2’ Declination- 10° 42.6’38° 47.8’19° 54.3’ AP - Latitude32° 16.5’32° 16.5’32° 16.5’ AP - Longitude- 80° 12.2’- 80° 12.2’ - 80° 12.2’ Observed altitude (Ho)44° 28.6’38° 39.5’ 8° 08.9’ Calculated altitude (Hc)44° 30.4’38° 48.4’ 7° 58.7’ Intercept (a-int)- 1.8’- 8.9’10.2’ Calculated Azimuth (Zn) 201.7°63.7°288.5°

References: The Nautical Almanac, Published Jointly by the U.S. Naval Observatory and the United Kingdom Hydrographic Office, The Nautical Almanac On-Line Version, A Short Guide to Celestial Navigation, Henning Umland, Celestial Navigation, David Burch, The Starpath School of Navigation, Longitude, Dava Sobel and William J. H. Andrews, Lunar Distances Explained, Arthur N. Pearson, Lunars.pdf CelNav, Program.xls and Guide.pdf, Ron Baker,

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